The document discusses solving quadratic equations by factoring. It provides examples of factoring quadratic expressions to find the solutions to the equations. These include using the zero product rule, factoring a common factor, and factoring a perfect square. It also provides two word problems involving consecutive integers and the Pythagorean theorem and shows how to set up and solve the quadratic equations derived from the word problems.
This will help you on how to solve quadratic equations by factoring.
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This will help you on how to solve quadratic equations by factoring.
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You will learn how to get the value of a, b and c given a quadratic equations.
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This powerpoint presentation discusses or talks about the topic or lesson Functions. It also discusses and explains the rules, steps and examples of Quadratic Functions.
This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
This powerpoint presentation discusses or talks about the topic or lesson Direct Variations. It also discusses and explains the rules, concepts, steps and examples of Direct Variations.
This will help you in factoring sum and difference of two cubes.
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You will learn how to get the value of a, b and c given a quadratic equations.
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https://tinyurl.com/y9muob6q
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https://tinyurl.com/ycjp8r7u
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This powerpoint presentation discusses or talks about the topic or lesson Functions. It also discusses and explains the rules, steps and examples of Quadratic Functions.
This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
This powerpoint presentation discusses or talks about the topic or lesson Direct Variations. It also discusses and explains the rules, concepts, steps and examples of Direct Variations.
This will help you in factoring sum and difference of two cubes.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
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https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Mathematics 9 Lesson 1-A: Solving Quadratic Equations by Completing the SquareJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson Solving Quadratic Equations using Completing the Square. It also discusses the steps in solving quadratic equations using the method of Completing the Square.
Mathematics 9 Lesson 1-C: Roots and Coefficients of Quadratic EquationsJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson Roots and Coefficients of Quadratic Equations. It also discusses and explains the rules, steps and examples of Roots and Coefficients of Quadratic Equations
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
1. Solving Quadratic Equations by Factoring Quadratic Equations are also known as Second Degree Equations because the highest power of the variable is 2. They may have zero, one or two solutions. There are several methods for solving them. This lesson involves those that can be solved by factoring.
2.
3. Then determine whether the expression on the left side of the equation can be factored. If so, then the product of those factors is 0, and since(factor1)• (factor2) = 0, then either factor1 = 0, or factor2 = 0.
4. Solve by Factoring Here is an example of an equation in Standard Form: y2 – 6y + 5 = 0 Can we factor the expression on the left side of the equation? Yes, we can express it as the product of two polynomial factors. Since the product of those two factors is zero, then according to the Zero Product Rule, one of the factors must be equal to zero. So we set each of the factors to zero and solve to determine the two possible solutions. y2 – 6y + 5 = 0 (y – 5) (y – 1) = 0 y – 5 = 0 OR y – 1 = 0 y = 5 y = 1
5. Check Substitute each of the solutions into the original equation for y = 5 y2 – 6y + 5 = 0 (5)2 – 6(5) + 5 = 0 25 – 30 + 5 = 0 -5 + 5 = 0 0 = 0 for y = 1 y2 – 6y + 5 = 0 (1)2– 6(1) + 5 = 0 1 – 6 + 5 = 0 -5 + 5 = 0 0 = 0 Both solutions are valid.
6. Solve by Factoring – common factor 9 x2 – 5 = 12x – 5 9x2 = 12x 9x2– 12x = 0 9x2– 12x = 0 3x(3x – 4) = 0 3x = 0 OR 3x - 4 = 0 x = 0 3x = 4 x = In this example, the equation is not in standard form, so the first step is to express the equation in standard form. To do so, we move the terms from the right to the left side of the equation. The constant term cancels out in our resulting equation. Next, we factor the expression on the right side of the equation. We find that we have a common factor of 3x in each of the terms of the expression. Using the zero product rule, we set each of the resulting factors equal to zero and solve to find the two solutions.
7. Check Substitute each of the solutions into the original equation for x = 0 9 x2 – 5 = 12x – 5 9(0)2 – 5 = 12(0) – 5 9(0) – 5 = 12(0) – 5 0 – 5 = 0 – 5 -5 = -5 for x = 9 x2 – 5 = 12x – 5 9( )2 – 5 = 12( ) – 5 9( ) – 5 = 12( ) – 5 16 – 5 = 16 - 5 11 = 11 Both solutions are valid.
8. Solve by Factoring – perfect square This equation is in standard form, so we need to determine whether the expression on the right side of the equation can be factored. First we find that there is a common factor of 3, so we factor it out. The resulting expression can be factored further. We recognize it as a perfect square. We can divide both sides of the equation by 3 and are left with two identical factors of (x + 2). According to the zero product rule, we know that x + 2 must be equal to 0, so we set it to zero and solve. 3x2 + 12x + 12 = 0 3(x2+ 4x + 4) = 0 3(x + 2)2 = 0 (x + 2)2 = 0 x + 2 = 0 x = -2
9. Check Substitute the solution into the original equation for x = -2 3x2 + 12x + 12 = 0 3(-2)2 + 12(-2) + 12 = 0 3(4) + 12 (-2) + 12 = 0 12 – 24 + 12 = 0 0 = 0 The solution is valid
10. Applications with Quadratic equations Consecutive Integer Problem We have three consecutive even integers. The sum of the first two integers is equal to one-fourth the product of the second and third. Find all possible solutions for the three integers. Using FFFSA method: FIND: We need to find three consecutive even integers. We can represent them as x, x +2 and x + 4 FACTS: The sum of the first two is represented by: x + (x + 2) The product of the second and the third is: (x + 2)(x + 4) One fourth of that value is: FORMULA: From the facts we can create an equation: x + x + 2 =
11. Consecutive Integer Problem SOLVE: x + x + 2 = 2x + 2 = combine like terms 4(2x + 2) = (x + 2)(x + 4) multiply both sides by 4 8x + 8 = x2 + 6x + 8 left: distributive property / right: FOIL 8x + 8 – 8 = x2 + 6x + 8 – 8 subtract 8 from both sides 8x = x2 + 6x simplify 0 = x2 + 6x – 8x subtract 8x from both sides 0 = x2 – 2x combine like terms x2 – 2x = 0 switch sides (symmetry principle) x(x - 2)= 0 factor out common factor of x x = 0 OR x – 2 = 0 Set factors equal to zero (zero x = 2 product rule) , and solve ANSWER: If x is 0, the second integer is x+2 = 0+2 = 2, the third integer is x+4 = 0+4 = 4 If x is 2, the second integer is x+2 = 2+2 = 4, the third integer is x+4 = 2+4 = 6 So we express the solutions as 0, 2, 4 OR 2, 4, 6
12. Check the SolutionConsecutive Integer Problem Let’s see if our answers fit into the original problem: We have three consecutive even integers. The sum of the first two integers is equal to one-fourth the product of the second and third. Find all possible solutions for the three integers. The solution 0, 2, 4: 0 + 2 = (2)(4) ÷ 4 2 = 8 ÷ 4 2 = 2 True The solution 2, 4, 6: 2 + 4 = (4)(6) ÷ 4 6 = 24 ÷ 4 6 = 6 True Both solutions are possible.
13. Applications with Quadratic equationsRight Triangle Problem The Pythagorean Theorem states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. It is represented by the formula: a2 + b2 = c2 Here is an example problem: The hypotenuse of a right triangle is 15 cm. and the larger leg is 12 cm. What is the length of the shorter leg? FIND: the length of the shorter leg. We will represent it as: a FACTS: b = 12, c = 15 FORMULA : a2 + b2 = c2 SUBSTITUTE: a2 + (12)2 = (15)2
14. Right Triangle Problem SOLVE: a2 + (12)2 = (15)2 a2 + 144 = 225 simplify a2 – 81 = 0 subtract 225 from both sides of the equation (a + 9) (a – 9) = 0 Since a2 and 81 are perfect squares, we can factor this as the difference of two squares a + 9 = 0 OR a – 9 = 0 Set each factor equal to 0 (Zero product rule) a = -9 a = 9 Solve. ANSWER: The two solutions are 9 and -9. If you check the solutions in the orignal equation, both with result in a true statement, but do they both make sense in the original problem? Remember, we are looking for the length of a side of a triangle. That length cannot be a negative number, so we can throw out the solution -9. So our final answer is that side a is equal to 9 cm.